Problem: Let $S$ be a piecewise-smooth, closed surface with an interior $V$. Suppose $F(x, y, z)$ is a continuously differentiable vector field. The surface $S$ spans through a regions with positive and negative $z$ and $y$. Does the divergence theorem necessarily apply to the region $V$, the boundary surface $S$, and the vector field $F$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Explanation: Assume we have a simple solid region $V$ oriented with outward normals, and it has a piecewise-smooth, closed boundary surface $S$. If $F$ is a continuously differentiable vector field in $\mathbb{R}^3$, then the divergence theorem says: $ \oiint_S F \cdot dS = \iiint_V \text{div}(F) \, dV$ The given surface, boundary, and vector field satisfy the conditions for the divergence theorem. It doesn't matter where $S$ is in space because we know that it's piecewise-smooth and closed. Therefore, we can apply the divergence theorem to them.